Question 8.15: Frequency response of a system from its pole-zero diagram Fi......

This can be factored into

\mathrm{H}(s)={\frac{(s+1-j4)(s+1+j4)}{(s+2-j10)(s+2+j10)}}.

So the poles and zeros of this transfer function are z_{1}=-1+j4,\ z_{2}=-1-j4 and  p_{1}=-2+j10,\;\;p_{2}=-2-j10 as illustrated in Figure 8.22.

Converting the transfer function to a frequency response,

\mathrm{H}(j\omega)={\frac{(j\omega+1-j4)(j\omega+1+j4)}{(j\omega+2-j10)(j\omega+2+j10)}}.

The magnitude of the frequency response at any particular frequency is the product of the numerator vector-magnitudes divided by the product of the denominator vector-magnitudes

|{H}(j\omega)\vert={\frac{|j\omega+1-j4||j\omega+1+j4|}{|j\omega+2-j10|j\omega+2+j10|}}.

The phase of the frequency response at any particular frequency is the sum of the numerator vector-angles minus the sum of the denominator vector-angles

\mathrm{\measuredangle}H(j\omega)=\measuredangle(j\omega+1-j\mathbf{4})+\measuredangle(j\omega+1+j\mathbf{4})-[\measuredangle(j\omega+2-j\mathbf{1}0)+\measuredangle (j\omega+2+j\mathbf{1}0)].

This transfer function has no poles or zeros on the \omega axis. Therefore its frequency response is neither zero nor infi nite at any real frequency. But the fi nite poles and fi nite zeros are near the real axis and, because of that proximity, will strongly infl uence the frequency response for real frequencies near those poles and zeros. For a real frequency \omega near the pole p_1 the denominator factor j\omega+2-j10 becomes very small and that makes the overall frequency response magnitude become very large. Conversely, for a real frequency \omega near the zero z_1 the numerator factor j{{\omega}}+1-j{\boldsymbol{4}}  becomes very small and that makes the overall frequency response magnitude become very small. So, not only does the frequency response magnitude go to zero at zeros and to infinity at poles, it becomes small near zeros and it becomes large near poles.

The frequency response magnitude and phase are illustrated in Figure 8.23.

Frequency response can be graphed using the MATLAB control toolbox command bode,
and pole-zero diagrams can be plotted using the MATLAB control toolbox command pzmap.